If a DNA from an ancient human, for example the cheddar man, is still intact, can you make an offspring today using that intact DNA?
You could, however the DNA is not that intact, there is a steady rate of loss of the genes...
mtDNA is degraded to an average length of 1 base pair after 6,830,000 years at −5 °C (see wiki: ancient dna). that's a speed of 1 base pair every 412 years. The cheddar caves are not frozen, I'd guess that the DNA lost one base pair every 10 or 20 years, so it has lost at least 400 or 800 base pairs. at 5 degrees, it may have lost 2 percent of it's informations, except that they sequenced it many times and reconstructed the original version.
The technology does exist to design synthetic chromosomes. https://www.nature.com/news/synthetic-yeast-chromosomes-help-probe-mysteries-of-evolution-1.21615
I figure that once science has a sure footing on genetic taboos, the do's and dont's, a company will one day try to give birth to neanderthals the cheddar specimen. There is too much scientific value for it to never be realized.
A species is generally defined as a group of organisms that can breed and produce fertile offspring. So, if you are extracting DNA from the an ancient human that is indeed the same species as us (homo sapiens), then by definition, you will be able to combine it with modern human DNA and generate offspring.
This is just stating the theory, the actual biology behind extracting that DNA and using it to fertilize offspring will be... messy.
scientifically yes, legally probably not as it would fall under human cloning/experimentation laws in most countries.
Artificial insemination with sperm with inserted genomes or artificial sperm already exists in mammals. Reconstructing genomes is possible and that together is all you really need.